bibkey: bharadwajrodgers2026largeprimes authors: Abhishek Bharadwaj; Brad Rodgers year: 2026 title: “Large prime factors of well-distributed sequences” doi: null url: https://arxiv.org/abs/2402.11884v4 claim: “Prime-factor statistics for weighted integer populations with positive polynomial index, positive level of distribution, and congruence uniformity; the ordinary unweighted value set of a fixed Fibonacci recurrence fails the positive-index hypothesis.” strata_touched: [] license: citation-only triage: anchor

Large prime-factor statistics and the FIB sampling contract

The source is arXiv:2402.11884v4, revised 9 April 2026; its first version appeared in February 2024. The following locators refer to v4. They record the definitions and main statements, not an independent audit of all proofs or a Lean verification.

Population and hypotheses

For nonnegative weights $a_m$, Section 1.2, printed pp.2–3, defines

$$N(x)=\sum _{m\le x}{a}_m,N_d(x)=\sum _{\begin{array}{c}m\le x\\ d\mid m\end{array}}{a}_m.$$

Condition (A) requires $N(x)=x^{\alpha +o(1)}$ for some $\alpha >0$. Condition (B) requires a level of distribution $\vartheta >0$: for each $0<c<\vartheta$ and every $A>0$,

$$\sum _{d\le x^c}|N_d(x)-g(d)N(x)|{\ll }_{c,A}\frac{N(x)}{(\log x{)}^A}.$$

The same condition requires a multiplicative $g(d)\in [0,1]$ satisfying ${\sum }_{p\le x}g(p)\log p=\log x+O(1)$ and $g(d)=O(C^{\Omega (d)}/d)$ for a constant $C\ge 1$. Condition (C), congruence uniformity, is

$$N_d(x)\ll \left(\frac{C^{\Omega (d)}}{d}N(x)+C^{\Omega (d)}\right)(\log x{)}^B(d\le x)$$

for fixed constants $B\ge 0$ and $C\ge 1$. The authors call the population $\sigma$ well-distributed when (A), (B), and (C) hold and $0<\sigma \le \min (\alpha ,\vartheta )$.

Exact statistical conclusions

Section 1.3 samples an integer $u$ with probability $a_u/N(x)$ for $1\le u\le x$. Its prime factors $p_1\ge p_2\ge \cdots$ are listed with multiplicity.

• Theorem 7, printed p.5, gives convergence of the normalized prime-factor process to Poisson–Dirichlet when the population is 1 well-distributed.
• Lemma 8, printed p.6, gives correlation convergence for a $\sigma$ well-distributed population only against compactly supported test functions in $\{(t_i):t_i>0,{\sum }_i{t}_i<\sigma \}$.
• Theorem 11, printed p.7, gives, for every $\epsilon >0$, $\limsup_{x\to\infty}\mathbb P(P^+(u)\ge u^{1-\varepsilon}) \ll\varepsilon$, with the constant depending on the population.

These are statements about the specified sampling law. Theorem 11 is an upper bound on the frequency of exceptionally large largest prime factors, not an upper bound on every selected integer’s divisor response.

Failed hypothesis for the ordinary fixed FIB value set

For a fixed nonzero nonnegative seed $(a,b)$, the FIB theory’s Binet formula, §199 gives $V_j=a{F}_{j+3}+b{F}_{j+4}=c_v{\varphi }^j+O_v(|\psi {|}^j)$, where $c_v>0$. Under the ordinary indicator $a_m=1_{\{m=V_j\text{ for some }j\ge 0\}}$, the values are eventually strictly increasing, and therefore

$$N(x)=\frac{\log x}{\log \varphi }+O_v(1),\frac{\log N(x)}{\log x}⟶0.$$

Condition (A) fails: this population has no positive polynomial index. Consequently the paper’s main results cannot be applied directly to that unweighted recurrence family, even before testing conditions (B) and (C). Taking a uniform random index $j\le J$ does not change the fact that the paper’s cutoff $x$ is the size of the integer value, not its index.

Reweighting the values or allowing seeds to vary changes the population; all three conditions and transport back to the desired actual candidates would then need separate proofs. No impossibility of every such alternative weighting is asserted here. Using all canonical FIB encodings instead recovers the ordinary integer population; its statistical theorem still does not control a specified extremal Robin candidate.

The remaining joint weight in FIB §233.5 is attached to one actual integer and its own divisors. Neither a distributional limit nor a density-one conclusion replaces that estimate, and this note introduces no new prime-factor statistical theorem.